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It is known that Dobrushin's ergodicity coefficient is one of the effective tools in the investigations of limiting behavior of Markov processes. Several interesting properties of the ergodicity coefficient of a positive mapping defined on…
Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results. (1) We show that constant SGD can be used as an…
We study a variance reduction strategy based on control variables for simulating the averaged macroscopic behavior of a stochastic slow-fast system. We assume that this averaged behavior can be written in terms of a few slow degrees of…
The past few decades have seen robust research on questions regarding the existence, form, and properties of stationary distributions of stochastically modeled reaction networks. When a stochastic model admits a stationary distribution an…
Ergodicity is a fundamental issue for a stochastic process. In this paper, we refine results on ergodicity for a general type of Markov chain to a specific type or the $GI/G/1$-type Markov chain, which has many interesting and important…
A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution…
We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the…
The inference of thermodynamic quantities from the description of an only partially accessible physical system is a central challenge in stochastic thermodynamics. A common approach is coarse-graining, which maps the dynamics of such a…
Dynamical processes can be transformed into graphs through a family of mappings called visibility algorithms, enabling the possibility of (i) making empirical data analysis and signal processing and (ii) characterising classes of dynamical…
The expansion of global production networks has raised many important questions about the interdependence among countries and how future changes in the world economy are likely to affect the countries' positioning in global value chains. We…
The stochastic processes underlying the growth and stability of biological and psychological systems reveal themselves when far from equilibrium. Far from equilibrium, nonergodicity reigns. Nonergodicity implies that the average outcome for…
In order to give quantitative estimates for approximating the ergodic limit, we investigate probabilistic limit behaviors of time-averaging estimators of numerical discretizations for a class of time-homogeneous Markov processes, by…
We prove that for a random walk on the real line whose increments have zero mean and are either integer-valued or spread out (i.e. the distributions of the steps of the walk are eventually non-singular), the Markov chain of overshoots above…
We propose a novel method to directly learn a stochastic transition operator whose repeated application provides generated samples. Traditional undirected graphical models approach this problem indirectly by learning a Markov chain model…
It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally…
We study a stochastic SIS (susceptible-infected-susceptible) epidemic dynamics on network, under the effect of a Markovian regime-switching. We first prove the existence of a unique global positive solution, and find a positive invariant…
We consider the problem of fitting variational posterior approximations using stochastic optimization methods. The performance of these approximations depends on (1) how well the variational family matches the true posterior…
We obtain the posterior distribution of a random process conditioned on observing the empirical frequencies of a finite sample path. We find under a rather broad assumption on the "dependence structure" of the process, {\em c.f.}…
Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy…
We consider an ordinary differential equation with a unique hyperbolic attractor at the origin, to which we add a small random perturbation. It is known that under general conditions, the solution of this stochastic differential equation…